We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hatπ : G \to \hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime field and if $V$ is a $k$-vector space, then, there is a natural isomorphism between $\hat{V}$ (for the underlying additive group structure) and the additive group of the double-dual $V^{**}$. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case.
A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based profinite group as a point in a Polish space. Then we study the complexity of isomorphism using the theory of Borel reducibility in descriptive set theory. For topologically finitely generated profinite groups this complexity is the same as the one of identity for reals. In general, it is the same as the complexity of isomorphism for countable graphs.
We consider pairs of finitely presented, residually finite groups $u:P\hookrightarrow Γ$. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not the associated map of profinite completions $\hat{u}: \widehat{P} \to \widehatΓ$ is an isomorphism. Nor do there exist algorithms that can decide whether $\hat{u}$ is surjective, or whether $\widehat{P}$ is isomorphic to $\widehatΓ$.
Let $S$ be a finitely generated pro-$p$ group. Let $\Emb(S)$ be the class of profinite groups $G$ that have $S$ as a Sylow subgroup, and such that $S$ intersects non-trivially with every non-trivial normal subgroup of $G$. In this paper, we investigate the question of whether or not $\Emb(S)$ has finitely many isomorphism classes. For instance, we give an example where $\Emb(S)$ contains an infinite ascending chain of soluble groups, and on the other hand show that $\Emb(S)$ contains only finitely many isomorphism classes in the case that $S$ is just infinite.
We fix a finitely presented group $Q$ and consider short exact sequences $1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$ induces a morphism of profinite completions $\hat N\to \hat G$. We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\subset G$, can determine whether or not $\hat P\to\hat G$ is an isomorphism.
In the first half of this paper, we outline the construction of a new class of abelian pro-$p$ groups, which covers all countably-based pro-$p$ groups. In the second half, we study them, and classify them up to topological isomorphism and abstract isomorphism. We use Ulm's classification of discrete countable p-groups, which are the Pontryagin duals of such pro-$p$ groups. It emerges that they are all abstractly isomorphic to Cartesian products of finite groups and $p$-adic integers. We have thus constructed uncountably many pairwise topologically non-isomorphic profinite groups abstractly iso…
We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions.
In this article we calculate the profinite genus of the fundamental group of a $n$-dimensional compact flat manifold $X$ with holonomy group of prime order. As consequence, we prove that if $n\leq 21$, then $X$ is determined among all $n$-dimensional compact flat manifolds by the profinite completion of its fundamental group. Furthermore, we characterize the isomorphism class of profinite completion of the fundamental group of $X$ in terms of the representation genus of its holonomy group.
Profinite genus of HNN-extensions with finite associated subgroups
Federal University of Viçosa（维索萨联邦大学）；ICT, Federal University of the Jequitinhonha and Mucuri Valleys（Jequitinhonha和Mucuri山谷联邦大学ICT学院）；Department of Mathematics, University of Brasilia（巴西利亚大学数学系）
Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size.
This article explores the interplay between the finite quotients of finitely generated residually finite groups and the concept of amenability. We construct a finitely generated, residually finite, amenable group $A$ and an uncountable family of finitely generated, residually finite non-amenable groups all of which are profinitely isomorphic to $A$. All of these groups are branch groups. Moreover, picking up Grothendieck's problem, the group $A$ embeds in these groups such that the inclusion induces an isomorphism of profinite completions. In addition, we review the concept of uniform amenabil…
We consider finitely presented,residually finite groups $G$ and finitely generated normal subgroups $A$ such that the inclusion $A\hookrightarrow G$ induces an isomorphism from the profinite completion of $A$ to a direct factor of the profinite completion of $G$. We explain why $A$ need not be a direct factor of a subgroup of finite index in $G$; indeed $G$ need not have a subgroup of finite index that splits as a non-trivial direct product. We prove that there is no algorithm that can determine whether $A$ is a direct factor of a subgroup of finite index in $G$.
For any orientable finite-volume hyperbolic 3-manifold, this paper proves that the profinite isomorphism type of the fundamental group uniquely determines the isomorphism type of the first integral cohomology, as marked with the Thurston norm and the fibered classes; moreover, up to finite ambiguity, the profinite isomorphism type determines the isomorphism type of the fundamental group, among the class of finitely generated 3-manifold groups.
Let $\C$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if $u:H\hookrightarrow G$ is a map of finitely generated residually $\C$ groups such that the induced map $\hat{u}:\hat{H}\rightarrow\hat{G}$ is a surjection of the pro-$\C$ completions, and $G$ has more than one end, then $H$ has the same number of ends as $G$. However if $G$ has one end the number of ends of $H$ may be larger; we observe cases where this occurs for $\C$ the class of finite $p$-groups. We produce a monomorphism of groups $u:H\hookrightarrow G$ such that: either $G$ is hyperbolic but not r…
A group is $\textit{finitely axiomatizable}$ (FA) in a class $\mathcal{C}$ if it can be determined up to isomorphism within $\mathcal{C}$ by a sentence in the first-order language of group theory. We show that profinite groups of various kinds are FA in the class of profinite groups. Reasons why certain groups cannot be FA are also discussed.
Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\to \hat{G}_p$ induces an isomorphism $H^\ast(\hat{G}_p,M)\to H^\ast(G,M)$ for any discrete $\hat{G}_p$-module $M$ of finite $p$-power order. For the general case, we prove that $G$ contains a normal subgroup $N$ of finite index such that the map $H^\ast(\hat{N}_p,M)\to H^\ast(N,M)$ is an isomorphism for any discrete $\hat{N}_p$-module …
We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index and every finitely presented group can be embedded in $U$. There is no algorithm that can determine whether or not a finitely presentable subgroup of a residually finite, biautomatic group is perfect. For every recursively presented abelian group $A$ there exists a pair of groups $i:P_A\hookrightarrow G_A$ such that $i$ induces an isomorphism of profinite c…
We consider pairs of finitely presented, residually finite groups $P\hookrightarrow\G$ for which the induced map of profinite completions $\hat P\to \hat\G$ is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to $\G$. We construct pairs for which the conjugacy problem in $\G$ can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does…
We show that a regular isomorphism of profinite completion of the fundamental groups of two 3-manifolds $N_1$ and $N_2$ induces an isometry of the Thurston norms and a bijection between the fibered classes. We study to what extent does the profinite completion of knot groups distinguish knots and show that it distinguishes each torus knot and the figure eight knot among all knots. We show also that it distinguishes between hyperbolic knots with cyclically commensurable complements under the assumption that their Alexander polynomials have at least one zero which is not a root of unity.
We study in detail the profinite group G arising as geometric étale iterated monodromy group of an arbitrary quadratic polynomial over a field of characteristic different from two. This is a self-similar closed subgroup of the group of automorphisms of a regular rooted binary tree. (When the base field is \C it is the closure of the finitely generated iterated monodromy group for the usual topology which is also often studied.) Among other things we prove that the conjugacy class and hence the isomorphism class of G depends only on the combinatorial type of the post-critical orbit of the polyn…
Assume $G$ is a polycyclic group and $ϕ:G\to G$ an endomorphism. Let $G\ast_ϕ$ be the ascending HNN extension of $G$ with respect to $ϕ$; that is, $G\ast_ϕ$ is given by the presentation $$G\ast_ϕ= < G, t \ |\ t^{-1}gt = ϕ(g)\ \{for all}\ g\in G >.$$ Furthermore, let $\hat{G\ast_ϕ}$ be the profinite completion of $G\ast_ϕ$. We prove that, for any finite discrete $\hat{G\ast_ϕ}$-module $A$, the map $H^*(\hat{G\ast_ϕ}, A)\to H^*(G\ast_ϕ,A)$ induced by the canonical map $G\ast_ϕ\to \hat{G\ast_ϕ}$ is an isomorphism.
We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound, which is not sharp because the relation is Borel.
The versal deformation ring R(G,V) of a mod p representation V of a profinite group G encodes all isomorphism classes of lifts of V to representations of G over complete local commutative Noetherian rings. We introduce a new technique for determining R(G,V) when G is finite which involves Brauer's generalized decomposition numbers.
Let $\{G_i :i\in\N\}$ be a family of finite Abelian groups. We say that a subgroup $G\leq \prod\limits_{i\in \N}G_i$ is \emph{order controllable} if for every $i\in \mathbb{N}$ there is $n_i\in \mathbb{N}$ such that for each $c\in G$, there exists $c_1\in G$ satisfying that $c_{1|[1,i]}=c_{|[1,i]}$, $supp (c_1)\subseteq [1,n_i]$, and order$(c_1)$ divides order$(c_{|[1,n_i]})$. In this paper we investigate the structure of order controllable subgroups. It is proved that every order controllable, profinite, abelian group contains a subset $\{g_n : n\in\N\}$ that topologically generates the group…
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
Let $KG$ denote the group algebra of the group $G$ over the field $K$ and let $U(KG)$ denote its group of units. Here without the use of a computer we give presentations for the unit groups of all group algebras $KG$, where the size of $KG$ is less than 1024. As a consequence we find the minimum counterexample to the Isomorphism Problem for group algebras.
Recognizing oriented knots from finite quotients with one marked element
Shiv Nadar Institution of Eminence Deemed to be University（希夫纳达尔卓越学院）；International Centre for Theoretical Sciences–Tata Institute of Fundamental Research（国际理论科学中心-塔塔基础研究所）
Let $S_\infty$ denote the topological group of permutations of the natural numbers. We study the complexity of the isomorphism relation on classes of closed subgroups $S_\infty$ in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup $G$ of $S_\infty$, the coarse group $\mathcal M(G)$ is the structure with domain the cosets of open subgroups of $G$, and a ternary relation $AB \sqsubseteq C$. If $G$ has only countably many open subgroups, then $\mathcal M(G)$ is a countable structure. Coarse groups form our main tool in studying such closed s…
A solution of the isomorphism problem is presented for the class of Coxeter groups W that have a finite set of Coxeter generators S such that the underlying graph of the presentation diagram of the system (W,S) has the property that every cycle of length at least four has a cord. As an application, we construct counterexamples to two main conjectures concerning the isomorphism problem for Coxeter groups.
Let $Γ(S)$ be the pure mapping class group of a connected orientable surface $S$ of negative Euler characteristic. For ${\mathscr C}$ a class of finite groups, let $\hatπ_1(S)^{\mathscr C}$ be the pro-${\mathscr C}$ completion of the fundamental group of $S$. The \emph{${\mathscr C}$-congruence completion $\checkΓ(S)^{\mathscr C}$ of $Γ(S)$} is the profinite completion induced by the embedding $Γ(S)\hookrightarrow{\operatorname{Out}}(\hatπ_1(S)^{\mathscr C})$. In this paper, we begin a systematic study of such completions for different ${\mathscr C}$. We show that the combinatorial structure o…
Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any…
Work on generalizations of the Cohen-Lenstra and Cohen-Martinet heuristics has drawn attention to probability measures on the space of isomorphism classes of profinite groups. As is common in probability theory, it would be desirable to know that these measures are determined by their moments, which in this context are the expected number of surjections to a fixed finite group. We show a wide class of measures, including those appearing in a recent paper of Liu, Wood, and Zurieck-Brown, have this property. The method is to work "locally" with groups that are extensions of a fixed group by a pr…
The Subgroup Isomorphism Problem for Integral Group Rings asks for which finite groups U it is true that if U is isomorphic to a subgroup of V(ZG), the group of normalized units of the integral group ring of the finite group G, it must be isomorphic to a subgroup of G. The smallest groups known not to satisfy this property are the counterexamples to the Isomorphism Problem constructed by M. Hertweck. However the only groups known to satisfy it are cyclic groups of prime power order and elementary-abelian p-groups of rank 2. We prove the Subgroup Isomorphism Problem for C_4 x C_2. Moreover we p…
